LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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zla_syrfsx_extended.f
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1*> \brief \b ZLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download ZLA_SYRFSX_EXTENDED + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zla_syrfsx_extended.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zla_syrfsx_extended.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zla_syrfsx_extended.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZLA_SYRFSX_EXTENDED( PREC_TYPE, UPLO, N, NRHS, A, LDA,
22* AF, LDAF, IPIV, COLEQU, C, B, LDB,
23* Y, LDY, BERR_OUT, N_NORMS,
24* ERR_BNDS_NORM, ERR_BNDS_COMP, RES,
25* AYB, DY, Y_TAIL, RCOND, ITHRESH,
26* RTHRESH, DZ_UB, IGNORE_CWISE,
27* INFO )
28*
29* .. Scalar Arguments ..
30* INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
31* $ N_NORMS, ITHRESH
32* CHARACTER UPLO
33* LOGICAL COLEQU, IGNORE_CWISE
34* DOUBLE PRECISION RTHRESH, DZ_UB
35* ..
36* .. Array Arguments ..
37* INTEGER IPIV( * )
38* COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
39* $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
40* DOUBLE PRECISION C( * ), AYB( * ), RCOND, BERR_OUT( * ),
41* $ ERR_BNDS_NORM( NRHS, * ),
42* $ ERR_BNDS_COMP( NRHS, * )
43* ..
44*
45*
46*> \par Purpose:
47* =============
48*>
49*> \verbatim
50*>
51*> ZLA_SYRFSX_EXTENDED improves the computed solution to a system of
52*> linear equations by performing extra-precise iterative refinement
53*> and provides error bounds and backward error estimates for the solution.
54*> This subroutine is called by ZSYRFSX to perform iterative refinement.
55*> In addition to normwise error bound, the code provides maximum
56*> componentwise error bound if possible. See comments for ERR_BNDS_NORM
57*> and ERR_BNDS_COMP for details of the error bounds. Note that this
58*> subroutine is only responsible for setting the second fields of
59*> ERR_BNDS_NORM and ERR_BNDS_COMP.
60*> \endverbatim
61*
62* Arguments:
63* ==========
64*
65*> \param[in] PREC_TYPE
66*> \verbatim
67*> PREC_TYPE is INTEGER
68*> Specifies the intermediate precision to be used in refinement.
69*> The value is defined by ILAPREC(P) where P is a CHARACTER and P
70*> = 'S': Single
71*> = 'D': Double
72*> = 'I': Indigenous
73*> = 'X' or 'E': Extra
74*> \endverbatim
75*>
76*> \param[in] UPLO
77*> \verbatim
78*> UPLO is CHARACTER*1
79*> = 'U': Upper triangle of A is stored;
80*> = 'L': Lower triangle of A is stored.
81*> \endverbatim
82*>
83*> \param[in] N
84*> \verbatim
85*> N is INTEGER
86*> The number of linear equations, i.e., the order of the
87*> matrix A. N >= 0.
88*> \endverbatim
89*>
90*> \param[in] NRHS
91*> \verbatim
92*> NRHS is INTEGER
93*> The number of right-hand-sides, i.e., the number of columns of the
94*> matrix B.
95*> \endverbatim
96*>
97*> \param[in] A
98*> \verbatim
99*> A is COMPLEX*16 array, dimension (LDA,N)
100*> On entry, the N-by-N matrix A.
101*> \endverbatim
102*>
103*> \param[in] LDA
104*> \verbatim
105*> LDA is INTEGER
106*> The leading dimension of the array A. LDA >= max(1,N).
107*> \endverbatim
108*>
109*> \param[in] AF
110*> \verbatim
111*> AF is COMPLEX*16 array, dimension (LDAF,N)
112*> The block diagonal matrix D and the multipliers used to
113*> obtain the factor U or L as computed by ZSYTRF.
114*> \endverbatim
115*>
116*> \param[in] LDAF
117*> \verbatim
118*> LDAF is INTEGER
119*> The leading dimension of the array AF. LDAF >= max(1,N).
120*> \endverbatim
121*>
122*> \param[in] IPIV
123*> \verbatim
124*> IPIV is INTEGER array, dimension (N)
125*> Details of the interchanges and the block structure of D
126*> as determined by ZSYTRF.
127*> \endverbatim
128*>
129*> \param[in] COLEQU
130*> \verbatim
131*> COLEQU is LOGICAL
132*> If .TRUE. then column equilibration was done to A before calling
133*> this routine. This is needed to compute the solution and error
134*> bounds correctly.
135*> \endverbatim
136*>
137*> \param[in] C
138*> \verbatim
139*> C is DOUBLE PRECISION array, dimension (N)
140*> The column scale factors for A. If COLEQU = .FALSE., C
141*> is not accessed. If C is input, each element of C should be a power
142*> of the radix to ensure a reliable solution and error estimates.
143*> Scaling by powers of the radix does not cause rounding errors unless
144*> the result underflows or overflows. Rounding errors during scaling
145*> lead to refining with a matrix that is not equivalent to the
146*> input matrix, producing error estimates that may not be
147*> reliable.
148*> \endverbatim
149*>
150*> \param[in] B
151*> \verbatim
152*> B is COMPLEX*16 array, dimension (LDB,NRHS)
153*> The right-hand-side matrix B.
154*> \endverbatim
155*>
156*> \param[in] LDB
157*> \verbatim
158*> LDB is INTEGER
159*> The leading dimension of the array B. LDB >= max(1,N).
160*> \endverbatim
161*>
162*> \param[in,out] Y
163*> \verbatim
164*> Y is COMPLEX*16 array, dimension (LDY,NRHS)
165*> On entry, the solution matrix X, as computed by ZSYTRS.
166*> On exit, the improved solution matrix Y.
167*> \endverbatim
168*>
169*> \param[in] LDY
170*> \verbatim
171*> LDY is INTEGER
172*> The leading dimension of the array Y. LDY >= max(1,N).
173*> \endverbatim
174*>
175*> \param[out] BERR_OUT
176*> \verbatim
177*> BERR_OUT is DOUBLE PRECISION array, dimension (NRHS)
178*> On exit, BERR_OUT(j) contains the componentwise relative backward
179*> error for right-hand-side j from the formula
180*> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
181*> where abs(Z) is the componentwise absolute value of the matrix
182*> or vector Z. This is computed by ZLA_LIN_BERR.
183*> \endverbatim
184*>
185*> \param[in] N_NORMS
186*> \verbatim
187*> N_NORMS is INTEGER
188*> Determines which error bounds to return (see ERR_BNDS_NORM
189*> and ERR_BNDS_COMP).
190*> If N_NORMS >= 1 return normwise error bounds.
191*> If N_NORMS >= 2 return componentwise error bounds.
192*> \endverbatim
193*>
194*> \param[in,out] ERR_BNDS_NORM
195*> \verbatim
196*> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
197*> For each right-hand side, this array contains information about
198*> various error bounds and condition numbers corresponding to the
199*> normwise relative error, which is defined as follows:
200*>
201*> Normwise relative error in the ith solution vector:
202*> max_j (abs(XTRUE(j,i) - X(j,i)))
203*> ------------------------------
204*> max_j abs(X(j,i))
205*>
206*> The array is indexed by the type of error information as described
207*> below. There currently are up to three pieces of information
208*> returned.
209*>
210*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
211*> right-hand side.
212*>
213*> The second index in ERR_BNDS_NORM(:,err) contains the following
214*> three fields:
215*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
216*> reciprocal condition number is less than the threshold
217*> sqrt(n) * slamch('Epsilon').
218*>
219*> err = 2 "Guaranteed" error bound: The estimated forward error,
220*> almost certainly within a factor of 10 of the true error
221*> so long as the next entry is greater than the threshold
222*> sqrt(n) * slamch('Epsilon'). This error bound should only
223*> be trusted if the previous boolean is true.
224*>
225*> err = 3 Reciprocal condition number: Estimated normwise
226*> reciprocal condition number. Compared with the threshold
227*> sqrt(n) * slamch('Epsilon') to determine if the error
228*> estimate is "guaranteed". These reciprocal condition
229*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
230*> appropriately scaled matrix Z.
231*> Let Z = S*A, where S scales each row by a power of the
232*> radix so all absolute row sums of Z are approximately 1.
233*>
234*> This subroutine is only responsible for setting the second field
235*> above.
236*> See Lapack Working Note 165 for further details and extra
237*> cautions.
238*> \endverbatim
239*>
240*> \param[in,out] ERR_BNDS_COMP
241*> \verbatim
242*> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
243*> For each right-hand side, this array contains information about
244*> various error bounds and condition numbers corresponding to the
245*> componentwise relative error, which is defined as follows:
246*>
247*> Componentwise relative error in the ith solution vector:
248*> abs(XTRUE(j,i) - X(j,i))
249*> max_j ----------------------
250*> abs(X(j,i))
251*>
252*> The array is indexed by the right-hand side i (on which the
253*> componentwise relative error depends), and the type of error
254*> information as described below. There currently are up to three
255*> pieces of information returned for each right-hand side. If
256*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
257*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
258*> the first (:,N_ERR_BNDS) entries are returned.
259*>
260*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
261*> right-hand side.
262*>
263*> The second index in ERR_BNDS_COMP(:,err) contains the following
264*> three fields:
265*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
266*> reciprocal condition number is less than the threshold
267*> sqrt(n) * slamch('Epsilon').
268*>
269*> err = 2 "Guaranteed" error bound: The estimated forward error,
270*> almost certainly within a factor of 10 of the true error
271*> so long as the next entry is greater than the threshold
272*> sqrt(n) * slamch('Epsilon'). This error bound should only
273*> be trusted if the previous boolean is true.
274*>
275*> err = 3 Reciprocal condition number: Estimated componentwise
276*> reciprocal condition number. Compared with the threshold
277*> sqrt(n) * slamch('Epsilon') to determine if the error
278*> estimate is "guaranteed". These reciprocal condition
279*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
280*> appropriately scaled matrix Z.
281*> Let Z = S*(A*diag(x)), where x is the solution for the
282*> current right-hand side and S scales each row of
283*> A*diag(x) by a power of the radix so all absolute row
284*> sums of Z are approximately 1.
285*>
286*> This subroutine is only responsible for setting the second field
287*> above.
288*> See Lapack Working Note 165 for further details and extra
289*> cautions.
290*> \endverbatim
291*>
292*> \param[in] RES
293*> \verbatim
294*> RES is COMPLEX*16 array, dimension (N)
295*> Workspace to hold the intermediate residual.
296*> \endverbatim
297*>
298*> \param[in] AYB
299*> \verbatim
300*> AYB is DOUBLE PRECISION array, dimension (N)
301*> Workspace.
302*> \endverbatim
303*>
304*> \param[in] DY
305*> \verbatim
306*> DY is COMPLEX*16 array, dimension (N)
307*> Workspace to hold the intermediate solution.
308*> \endverbatim
309*>
310*> \param[in] Y_TAIL
311*> \verbatim
312*> Y_TAIL is COMPLEX*16 array, dimension (N)
313*> Workspace to hold the trailing bits of the intermediate solution.
314*> \endverbatim
315*>
316*> \param[in] RCOND
317*> \verbatim
318*> RCOND is DOUBLE PRECISION
319*> Reciprocal scaled condition number. This is an estimate of the
320*> reciprocal Skeel condition number of the matrix A after
321*> equilibration (if done). If this is less than the machine
322*> precision (in particular, if it is zero), the matrix is singular
323*> to working precision. Note that the error may still be small even
324*> if this number is very small and the matrix appears ill-
325*> conditioned.
326*> \endverbatim
327*>
328*> \param[in] ITHRESH
329*> \verbatim
330*> ITHRESH is INTEGER
331*> The maximum number of residual computations allowed for
332*> refinement. The default is 10. For 'aggressive' set to 100 to
333*> permit convergence using approximate factorizations or
334*> factorizations other than LU. If the factorization uses a
335*> technique other than Gaussian elimination, the guarantees in
336*> ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy.
337*> \endverbatim
338*>
339*> \param[in] RTHRESH
340*> \verbatim
341*> RTHRESH is DOUBLE PRECISION
342*> Determines when to stop refinement if the error estimate stops
343*> decreasing. Refinement will stop when the next solution no longer
344*> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
345*> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
346*> default value is 0.5. For 'aggressive' set to 0.9 to permit
347*> convergence on extremely ill-conditioned matrices. See LAWN 165
348*> for more details.
349*> \endverbatim
350*>
351*> \param[in] DZ_UB
352*> \verbatim
353*> DZ_UB is DOUBLE PRECISION
354*> Determines when to start considering componentwise convergence.
355*> Componentwise convergence is only considered after each component
356*> of the solution Y is stable, which we define as the relative
357*> change in each component being less than DZ_UB. The default value
358*> is 0.25, requiring the first bit to be stable. See LAWN 165 for
359*> more details.
360*> \endverbatim
361*>
362*> \param[in] IGNORE_CWISE
363*> \verbatim
364*> IGNORE_CWISE is LOGICAL
365*> If .TRUE. then ignore componentwise convergence. Default value
366*> is .FALSE..
367*> \endverbatim
368*>
369*> \param[out] INFO
370*> \verbatim
371*> INFO is INTEGER
372*> = 0: Successful exit.
373*> < 0: if INFO = -i, the ith argument to ZLA_HERFSX_EXTENDED had an illegal
374*> value
375*> \endverbatim
376*
377* Authors:
378* ========
379*
380*> \author Univ. of Tennessee
381*> \author Univ. of California Berkeley
382*> \author Univ. of Colorado Denver
383*> \author NAG Ltd.
384*
385*> \ingroup complex16SYcomputational
386*
387* =====================================================================
388 SUBROUTINE zla_syrfsx_extended( PREC_TYPE, UPLO, N, NRHS, A, LDA,
389 $ AF, LDAF, IPIV, COLEQU, C, B, LDB,
390 $ Y, LDY, BERR_OUT, N_NORMS,
391 $ ERR_BNDS_NORM, ERR_BNDS_COMP, RES,
392 $ AYB, DY, Y_TAIL, RCOND, ITHRESH,
393 $ RTHRESH, DZ_UB, IGNORE_CWISE,
394 $ INFO )
395*
396* -- LAPACK computational routine --
397* -- LAPACK is a software package provided by Univ. of Tennessee, --
398* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
399*
400* .. Scalar Arguments ..
401 INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
402 $ N_NORMS, ITHRESH
403 CHARACTER UPLO
404 LOGICAL COLEQU, IGNORE_CWISE
405 DOUBLE PRECISION RTHRESH, DZ_UB
406* ..
407* .. Array Arguments ..
408 INTEGER IPIV( * )
409 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
410 $ y( ldy, * ), res( * ), dy( * ), y_tail( * )
411 DOUBLE PRECISION C( * ), AYB( * ), RCOND, BERR_OUT( * ),
412 $ ERR_BNDS_NORM( NRHS, * ),
413 $ ERR_BNDS_COMP( NRHS, * )
414* ..
415*
416* =====================================================================
417*
418* .. Local Scalars ..
419 INTEGER UPLO2, CNT, I, J, X_STATE, Z_STATE,
420 $ Y_PREC_STATE
421 DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
422 $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
423 $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
424 $ EPS, HUGEVAL, INCR_THRESH
425 LOGICAL INCR_PREC, UPPER
426 COMPLEX*16 ZDUM
427* ..
428* .. Parameters ..
429 INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
430 $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
431 $ extra_y
432 parameter( unstable_state = 0, working_state = 1,
433 $ conv_state = 2, noprog_state = 3 )
434 parameter( base_residual = 0, extra_residual = 1,
435 $ extra_y = 2 )
436 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
437 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
438 INTEGER CMP_ERR_I, PIV_GROWTH_I
439 PARAMETER ( FINAL_NRM_ERR_I = 1, final_cmp_err_i = 2,
440 $ berr_i = 3 )
441 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
442 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
443 $ piv_growth_i = 9 )
444 INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
445 $ la_linrx_cwise_i
446 parameter( la_linrx_itref_i = 1,
447 $ la_linrx_ithresh_i = 2 )
448 parameter( la_linrx_cwise_i = 3 )
449 INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
450 $ la_linrx_rcond_i
451 parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
452 parameter( la_linrx_rcond_i = 3 )
453* ..
454* .. External Functions ..
455 LOGICAL LSAME
456 EXTERNAL ILAUPLO
457 INTEGER ILAUPLO
458* ..
459* .. External Subroutines ..
460 EXTERNAL zaxpy, zcopy, zsytrs, zsymv, blas_zsymv_x,
461 $ blas_zsymv2_x, zla_syamv, zla_wwaddw,
463 DOUBLE PRECISION DLAMCH
464* ..
465* .. Intrinsic Functions ..
466 INTRINSIC abs, real, dimag, max, min
467* ..
468* .. Statement Functions ..
469 DOUBLE PRECISION CABS1
470* ..
471* .. Statement Function Definitions ..
472 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
473* ..
474* .. Executable Statements ..
475*
476 info = 0
477 upper = lsame( uplo, 'U' )
478 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
479 info = -2
480 ELSE IF( n.LT.0 ) THEN
481 info = -3
482 ELSE IF( nrhs.LT.0 ) THEN
483 info = -4
484 ELSE IF( lda.LT.max( 1, n ) ) THEN
485 info = -6
486 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
487 info = -8
488 ELSE IF( ldb.LT.max( 1, n ) ) THEN
489 info = -13
490 ELSE IF( ldy.LT.max( 1, n ) ) THEN
491 info = -15
492 END IF
493 IF( info.NE.0 ) THEN
494 CALL xerbla( 'ZLA_HERFSX_EXTENDED', -info )
495 RETURN
496 END IF
497 eps = dlamch( 'Epsilon' )
498 hugeval = dlamch( 'Overflow' )
499* Force HUGEVAL to Inf
500 hugeval = hugeval * hugeval
501* Using HUGEVAL may lead to spurious underflows.
502 incr_thresh = dble( n ) * eps
503
504 IF ( lsame( uplo, 'L' ) ) THEN
505 uplo2 = ilauplo( 'L' )
506 ELSE
507 uplo2 = ilauplo( 'U' )
508 ENDIF
509
510 DO j = 1, nrhs
511 y_prec_state = extra_residual
512 IF ( y_prec_state .EQ. extra_y ) THEN
513 DO i = 1, n
514 y_tail( i ) = 0.0d+0
515 END DO
516 END IF
517
518 dxrat = 0.0d+0
519 dxratmax = 0.0d+0
520 dzrat = 0.0d+0
521 dzratmax = 0.0d+0
522 final_dx_x = hugeval
523 final_dz_z = hugeval
524 prevnormdx = hugeval
525 prev_dz_z = hugeval
526 dz_z = hugeval
527 dx_x = hugeval
528
529 x_state = working_state
530 z_state = unstable_state
531 incr_prec = .false.
532
533 DO cnt = 1, ithresh
534*
535* Compute residual RES = B_s - op(A_s) * Y,
536* op(A) = A, A**T, or A**H depending on TRANS (and type).
537*
538 CALL zcopy( n, b( 1, j ), 1, res, 1 )
539 IF ( y_prec_state .EQ. base_residual ) THEN
540 CALL zsymv( uplo, n, dcmplx(-1.0d+0), a, lda, y(1,j), 1,
541 $ dcmplx(1.0d+0), res, 1 )
542 ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
543 CALL blas_zsymv_x( uplo2, n, dcmplx(-1.0d+0), a, lda,
544 $ y( 1, j ), 1, dcmplx(1.0d+0), res, 1, prec_type )
545 ELSE
546 CALL blas_zsymv2_x(uplo2, n, dcmplx(-1.0d+0), a, lda,
547 $ y(1, j), y_tail, 1, dcmplx(1.0d+0), res, 1,
548 $ prec_type)
549 END IF
550
551! XXX: RES is no longer needed.
552 CALL zcopy( n, res, 1, dy, 1 )
553 CALL zsytrs( uplo, n, 1, af, ldaf, ipiv, dy, n, info )
554*
555* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
556*
557 normx = 0.0d+0
558 normy = 0.0d+0
559 normdx = 0.0d+0
560 dz_z = 0.0d+0
561 ymin = hugeval
562
563 DO i = 1, n
564 yk = cabs1( y( i, j ) )
565 dyk = cabs1( dy( i ) )
566
567 IF ( yk .NE. 0.0d+0 ) THEN
568 dz_z = max( dz_z, dyk / yk )
569 ELSE IF ( dyk .NE. 0.0d+0 ) THEN
570 dz_z = hugeval
571 END IF
572
573 ymin = min( ymin, yk )
574
575 normy = max( normy, yk )
576
577 IF ( colequ ) THEN
578 normx = max( normx, yk * c( i ) )
579 normdx = max( normdx, dyk * c( i ) )
580 ELSE
581 normx = normy
582 normdx = max( normdx, dyk )
583 END IF
584 END DO
585
586 IF ( normx .NE. 0.0d+0 ) THEN
587 dx_x = normdx / normx
588 ELSE IF ( normdx .EQ. 0.0d+0 ) THEN
589 dx_x = 0.0d+0
590 ELSE
591 dx_x = hugeval
592 END IF
593
594 dxrat = normdx / prevnormdx
595 dzrat = dz_z / prev_dz_z
596*
597* Check termination criteria.
598*
599 IF ( ymin*rcond .LT. incr_thresh*normy
600 $ .AND. y_prec_state .LT. extra_y )
601 $ incr_prec = .true.
602
603 IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
604 $ x_state = working_state
605 IF ( x_state .EQ. working_state ) THEN
606 IF ( dx_x .LE. eps ) THEN
607 x_state = conv_state
608 ELSE IF ( dxrat .GT. rthresh ) THEN
609 IF ( y_prec_state .NE. extra_y ) THEN
610 incr_prec = .true.
611 ELSE
612 x_state = noprog_state
613 END IF
614 ELSE
615 IF (dxrat .GT. dxratmax) dxratmax = dxrat
616 END IF
617 IF ( x_state .GT. working_state ) final_dx_x = dx_x
618 END IF
619
620 IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
621 $ z_state = working_state
622 IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
623 $ z_state = working_state
624 IF ( z_state .EQ. working_state ) THEN
625 IF ( dz_z .LE. eps ) THEN
626 z_state = conv_state
627 ELSE IF ( dz_z .GT. dz_ub ) THEN
628 z_state = unstable_state
629 dzratmax = 0.0d+0
630 final_dz_z = hugeval
631 ELSE IF ( dzrat .GT. rthresh ) THEN
632 IF ( y_prec_state .NE. extra_y ) THEN
633 incr_prec = .true.
634 ELSE
635 z_state = noprog_state
636 END IF
637 ELSE
638 IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
639 END IF
640 IF ( z_state .GT. working_state ) final_dz_z = dz_z
641 END IF
642
643 IF ( x_state.NE.working_state.AND.
644 $ ( ignore_cwise.OR.z_state.NE.working_state ) )
645 $ GOTO 666
646
647 IF ( incr_prec ) THEN
648 incr_prec = .false.
649 y_prec_state = y_prec_state + 1
650 DO i = 1, n
651 y_tail( i ) = 0.0d+0
652 END DO
653 END IF
654
655 prevnormdx = normdx
656 prev_dz_z = dz_z
657*
658* Update soluton.
659*
660 IF ( y_prec_state .LT. extra_y ) THEN
661 CALL zaxpy( n, dcmplx(1.0d+0), dy, 1, y(1,j), 1 )
662 ELSE
663 CALL zla_wwaddw( n, y(1,j), y_tail, dy )
664 END IF
665
666 END DO
667* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
668 666 CONTINUE
669*
670* Set final_* when cnt hits ithresh.
671*
672 IF ( x_state .EQ. working_state ) final_dx_x = dx_x
673 IF ( z_state .EQ. working_state ) final_dz_z = dz_z
674*
675* Compute error bounds.
676*
677 IF ( n_norms .GE. 1 ) THEN
678 err_bnds_norm( j, la_linrx_err_i ) =
679 $ final_dx_x / (1 - dxratmax)
680 END IF
681 IF ( n_norms .GE. 2 ) THEN
682 err_bnds_comp( j, la_linrx_err_i ) =
683 $ final_dz_z / (1 - dzratmax)
684 END IF
685*
686* Compute componentwise relative backward error from formula
687* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
688* where abs(Z) is the componentwise absolute value of the matrix
689* or vector Z.
690*
691* Compute residual RES = B_s - op(A_s) * Y,
692* op(A) = A, A**T, or A**H depending on TRANS (and type).
693*
694 CALL zcopy( n, b( 1, j ), 1, res, 1 )
695 CALL zsymv( uplo, n, dcmplx(-1.0d+0), a, lda, y(1,j), 1,
696 $ dcmplx(1.0d+0), res, 1 )
697
698 DO i = 1, n
699 ayb( i ) = cabs1( b( i, j ) )
700 END DO
701*
702* Compute abs(op(A_s))*abs(Y) + abs(B_s).
703*
704 CALL zla_syamv ( uplo2, n, 1.0d+0,
705 $ a, lda, y(1, j), 1, 1.0d+0, ayb, 1 )
706
707 CALL zla_lin_berr ( n, n, 1, res, ayb, berr_out( j ) )
708*
709* End of loop for each RHS.
710*
711 END DO
712*
713 RETURN
714*
715* End of ZLA_SYRFSX_EXTENDED
716*
717 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zla_wwaddw(N, X, Y, W)
ZLA_WWADDW adds a vector into a doubled-single vector.
Definition: zla_wwaddw.f:81
subroutine zla_lin_berr(N, NZ, NRHS, RES, AYB, BERR)
ZLA_LIN_BERR computes a component-wise relative backward error.
Definition: zla_lin_berr.f:101
subroutine zsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZSYMV computes a matrix-vector product for a complex symmetric matrix.
Definition: zsymv.f:157
subroutine zla_syrfsx_extended(PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
ZLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric inde...
subroutine zsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZSYTRS
Definition: zsytrs.f:120
subroutine zla_syamv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bou...
Definition: zla_syamv.f:179